Is there a way to make a render animation take its information from texture mode?
You can trigger what is called an OpenGL Render in any of the Viewport Shading modes (solid, textured, wire, matcap,...). Switch to the mode you want first (in your case Textured Mode, make sure the output settings in the render panel are set correctly, and use the Render OpenGL Animation or Render OpenGL Image button from the 3D Viewport:



That should only take a few seconds to complete on short animations. If you need to tweak the quality, you can use the menu in the Info header bar:



You can set Anti-Aliasing and the background transparency directly from there.